Analysis of Tomographic Pumping Tests with Regularized Inversion Geoffrey C. Bohling Kansas Geological Survey SIAM Geosciences Conference Santa Fe, NM,

Analysis of Tomographic Pumping Tests with Regularized Inversion

Geoffrey C. BohlingKansas Geological SurveySIAM Geosciences ConferenceSanta Fe, NM, 22 March 2007

Santa Fe, 22 March 2007

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Hydraulic Tomography

Simultaneous analysis of multiple tests (or stresses) with multiple observation points

Information from multiple flowpaths helps reduce nonuniqueness

But still the same inverse problem we have been dealing with for decades


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Forward and Inverse Modeling Forward Problem: d = G(m)

Approximate: No model represents true mapping from parameter space (m) to data space (d)

Nonunique: BIG m small d Inverse Problem: m = G-1(d)

Effective: Estimated parameters are always “effective” (contingent on approximate model)

Unstable: small d BIG m


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Regularizing the Inverse Problem Groundwater flow models potentially have a

very large number of parameters Uncontrolled inversion with many parameters

can match almost anything, most likely with wildly varying parameter estimates

Regularization restricts variation of parameters to try to keep them plausible

Regularization by zonation is traditional approach in groundwater modeling


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Tikhonov Regularization (Damped L.S.) 2

2

22

2)(minˆ refmmLdmGm

m

Allow a large number of parameters (vector, m), but regularize by penalizing deviations from reference model, mref

Balancing residual norm (sum of squared residuals) against model norm (squared deviations from reference model)

Increasing regularization parameter, , gives “smoother” solution

Reduces instability of inversion and avoids overfitting data

L is identity matrix for zeroth-order regularization, numerical Laplacian for second-order regularization

Plot of model norm versus residual norm with varying is an “L-curve” – used in selecting appropriate level of regularization


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Field Site (GEMS)

Highly permeable alluvial aquifer (K ~ 1.5x10-3 m/s)

Many experiments over past 19 years

Induced gradient tracer test (GEMSTRAC1) in 1994

Hydraulic tomography experiments in 2002

Various direct push tests over past 7 years


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Field Site Stratigraphy

From Butler, 2005, in Hydrogeophysics (Rubin and Hubbard, eds.), 23-58


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Tomographic Pumping Tests


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Drawdowns from Gems4S Tests

From Bohling et al., 2007, Water Resources Research, in press


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Analysis Approach

Forward simulation with 2D radial-vertical flow model in Matlab Vertical “wedge” emanating from pumping well Common 10 x 14 Cartesian grid of lnK values mapped into

radial grid for each pumping well Inverse analysis with Matlab nonlinear least squares

function, lsqnonlin Fitting parameters are Cartesian grid lnK values Regularization relative to uniform lnK (K = 1.5 x 10-3 m/s)

model for varying values of Steady-shape analysis


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Parallel Synthetic Experiments For guidance, tomographic pumping tests simulated

in Modflow using synthetic aquifer Vertical lnK variogram for synthetic aquifer derived

from GEMSTRAC1 lnK profile Vertical profile includes fining upward trend and

periodic (cyclic) component Large horizontal range (61 m) yields “imperfectly

layered” aquifer K values range from 4.9 x 10-5 m/s (silty to clean

sand) to 1.7 x 10-2 m/s (clean sand to gravel) with a geometric mean of 1.2 x 10-3 m/s


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Synthetic Aquifer (81 x 49 x 70)


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Four Grids


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L-curves, Real and Synthetic Tests


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Synthetic Results


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Model Norm Relative to “Truth”


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Real Results


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Transient Fit, Gems4SUsing K field for = 0.025 with Ss = 3x10-5 m-1


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Transient Fit, Gems4NUsing K field for = 0.025 with Ss = 3x10-5 m-1


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Well Locations


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Comparison to Other Estimates


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Conclusions

Synthetic results show that steady-shape radial analysis of tests captures salient features of K field, but also indicate “effective” nature of fits

For real tests, pattern of estimated K probably reasonable, although range of estimated values may be too wide

A lot of effort to characterize a 10 m x 10 m section of aquifer; perhaps not feasible for routine aquifer characterization studies

Should be valuable for detailed characterization at research sites


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Acknowledgment s

Field effort led by Jim Butler with support from John Healey, Greg Davis, and Sam Cain

Support from NSF grant 9903103 and KGS Applied Geohydrology Summer Research Assistantship Program


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Regularizing w.r.t. Stochastic PriorsSecond-order regularization – asking for smooth variations from prior model

Fairly strong regularization here (α = 0.1)

Best 5 fits of 50

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Analysis of Tomographic Pumping Tests with Regularized Inversion Geoffrey C. Bohling Kansas Geological Survey SIAM Geosciences Conference Santa Fe, NM,